Nonstationary stochastic gold-mining: A time-sequential tactical-allocation problem

This paper presents an extension of gold-mining problems formulated in earlier work by R. Bellman and J. Kadane. Bellman assumes there are two gold mines labeled A and B, respectively, each with a known initial amount of gold. There is one delicate gold-mining machine which can be used to excavate one mine per day. Associated with mine A is a known constant return rate and a known constant probability of breakdown. There is also a return rate and probability of breakdown for mine B. Bellman solves the problem of finding a sequential decision procedure to maximize the expected amount of gold obtained before breakdown of the machine. Kadane extends the problem by assuming that there are several mines and that there are sequences of constants such that the jth constant for each mine represents the return rate for the jth excavation of that mine. He also assumes that the probability of breakdown during the jth excavation of a mine depends on j. We extend these results by assuming that the return rates are random variables with known joint distribution and by allowing the probability of breakdown to be a function of previous observations on the return rates. We show that under certain regularity conditions on the joint distributions of the random variables, the optimal policy is: at each stage always select a mine which has maximal conditional expected return per unit risk. This gold-mining problem is also a formulation of the problem of time-sequential tactical allocation of bombers to targets. Several examples illustrating these results are presented.     

Optimum sequential search with discrete locations and random acceptance errors

Much work has been done in search theory; however, very little effort has occurred where an object's presence at a location can be accepted when no object is present there. The case analyzed is of this type. The number of locations is finite, a single object is stationary at one location, and only one location is observed each step of the search. The object's location has a known prior probability distribution. Also known are the conditional probability of acceptance given the object's absence (small) and the conditional probability of rejection given the object's presence (not too large); these Probabilities remain fixed for all searching and locations. The class of sequential search policies which terminate the search at the first acceptance is assumed. A single two-part optimization criterion is considered. The search sequence is found which (i) minimizes the probability of obtaining n rejections in the first n steps for all n, and (ii) maximizes the probability that the first acceptance occurs within the first n steps and occurs at the object's location for all n. The optimum sequential search policy specifies that the next location observed is one with the largest posterior probability of the object's presence (evaluated after each step from Bayes Rule) and that the object is at the first location where acceptance occurs. Placement at the first acceptance seems appropriate when the conditional probability of acceptance given the object's absence is sufficiently small. Search always terminates (with probability one). Optimum truncated sequential policies are also considered. Methods are given for evaluating some pertinent properties and for investigating the possibility that no object occurs at any location.     

The discrete search problem and the construction of optimal allocations

Suppose one object is hidden in the k-th of n boxes with probability p(k). The boxes are to be searched sequentially. Associated with the j-th search of box k is a cost c(j,k) and a conditional probability q(j,k) that the first j - 1 searches of box k are unsuccessful while the j-th search is successful given that the object is hidden in box k. The problem is to maximize the probability that we find the object if we are not allowed to offer more than L for the search. We prove the existence of an optimal allocation of the search effort L and state an algorithm for the construction of an optimal allocation. Finally, we discuss some problems concerning the complexity of our problem.     

Technical note: Find a hidden “treasure”

This paper deals with a two searchers game and it investigates the problem of how the possibility of finding a hidden object simultaneously by players influences their behavior. Namely, we consider the following two‐sided allocation non‐zero‐sum game on an integer interval [1,n]. Two teams (Player 1 and 2) want to find an immobile object (say, a treasure) hidden at one of n points. Each point i ∈ [1,n] is characterized by a detection parameter λ_i (μ_i) for Player 1 (Player 2) such that p_i(1 ? exp(?λ_ix_i)) (p_i(1 ? exp(?μ_iy_i))) is the probability that Player 1 (Player 2) discovers the hidden object with amount of search effort x_i (y_i) applied at point i where p_i ∈ (0,1) is the probability that the object is hidden at point i. Player 1 (Player 2) undertakes the search by allocating the total amount of effort X(Y). The payoff for Player 1 (Player 2) is 1 if he detects the object but his opponent does not. If both players detect the object they can share it proportionally and even can pay some share to an umpire who takes care that the players do not cheat each other, namely Player 1 gets q₁ and Player 2 gets q₂ where q₁ + q₂ ≤ 1. The Nash equilibrium of this game is found and numerical examples are given. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A markov chain version of the secretary problem

This paper deals with the Secretary Problem where n secretaries are interviewed sequentially and the best k must be hired. The values of the secretaries are observed as they are interviewed, but beforehand only the distributions of these values are known. Furthermore, the distributions of two successive secretaries' values are governed by a Markov chain. Optimal hiring policies for finite n and limiting optimal policies as k and n approach infinity are obtained.     

Consecutive k‐within‐m‐out‐of‐n:F system with exchangeable components

As a generalization of k‐out‐of‐n:F and consecutive k‐out‐of‐n:F systems, the consecutive k‐within‐m‐out‐of‐n:F system consists of n linearly ordered components such that the system fails iff there are m consecutive components which include among them at least k failed components. In this article, the reliability properties of consecutive k‐within‐m‐out‐of‐n:F systems with exchangeable components are studied. The bounds and approximations for the survival function are provided. A Monte Carlo estimator of system signature is obtained and used to approximate survival function. The results are illustrated and numerics are provided for an exchangeable multivariate Pareto distribution. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

考虑修理工可变休假的n中取k系统可靠性模型

针对装备保障中维修调度对装备训练及可靠性的影响,将支队级修理所保障多艘舰船维修工作的情况抽象为单一维修台保障多个系统的维修力量调度分配,引入修理工可变休假策略对其进行描述,以装备结构中常见的n中取k系统为研究对象,针对以往研究利用指数分布等典型分布导致模型约束条件过于严格的问题,利用连续Phase-type分布描述了系统相关随机变量,构建系统可靠性解析模型,通过算例验证了模型适用性,模拟分析了修理工有无休假、修理工休假速率等相关因子对系统运行指标产生的各种影响。算例结果表明,该可靠性模型可以有效复现维修力量调度对n中取k系统可靠性的影响,可为修理工休假次数的合理安排、系统部件数量的优化配置提供理论基础和实践参考。     

Bayes sequential strategies for crossing a field containing absorption points

A sequential decision problem is considered in which N particles have to cross a given field. Two alternative crossing paths are available. An unknown number of absorption points J₁ and J₂ are planted at each of the crossing paths. The bivariate prior distribution of (J₁,J₂) is given. If a particle passes close to an absorption point it may survive with probability s, 0 < s < 1. If a particle is absorbed, both the particle and the absorption point are ruined. There is no replacement of ruined absorption points. All absorption points act independently. The particles crciss the field in a consecutive order, and a crossing path can be chosen for each particle. The objective is to maximize the expected number of survivors. The Bayes sequential procedure is characterized. The csmditions under which the Bayes strategy is determined by the maximal posterior survival probabilities are specified.     

On the first time a separately maintained parallel system has been down for a fixed time

Consider a system consisting of n separately maintained independent components where the components alternate between intervals in which they are “up” and in which they are “down”. When the i^th component goes up [down] then, independent of the past, it remains up [down] for a random length of time, having distribution F_i[G_i], and then goes down [up]. We say that component i is failed at time t if it has been “down” at all time points s ?[t-A.t]: otherwise it is said to be working. Thus, a component is failed if it is down and has been down for the previous A time units. Assuming that all components initially start “up,” let T denote the first time they are all failed, at which point we say the system is failed. We obtain the moment-generating function of T when n = l, for general F and G, thus generalizing previous results which assumed that at least one of these distributions be exponential. In addition, we present a condition under which T is an NBU (new better than used) random variable. Finally we assume that all the up and down distributions F_i and G_i i = l,….n, are exponential, and we obtain an exact expression for E(T) for general n; in addition we obtain bounds for all higher moments of T by showing that T is NBU.     

Isotonic median regression for orders representable by rooted trees

The isotonic median regression problem arising in statistics is as follows. We are given m observations falling into n sets, the ith set containing m_i observations. The problem requires the determination of n real numbers, the ith being the value “fitted” to each observation in the ith set. These n numbers chosen must satisfy certain (total or partial) order requirements and minimize the distance between the vectors of observed and fitted values in the l₁ norm. We present a simple algorithm, of time complexity O(mn), for calculating isotonic median regression for orders representable by rooted trees. We believe that this algorithm is the best currently available for this problem. The algorithm is validated by a linear programming approach which provides additional insight.     

Signature based analysis of m‐Consecutive‐k‐out‐of‐n: F systems with exchangeable components

In this article, we study reliability properties of m‐consecutive‐k‐out‐of‐n: F systems with exchangeable components. We deduce exact formulae and recurrence relations for the signature of the system. Closed form expressions for the survival function and the lifetime distribution as a mixture of the distribution of order statistics are established as well. These representations facilitate the computation of several reliability characteristics of the system for a given exchangeable joint distribution or survival function. Finally, we provide signature‐based stochastic ordering results for the system's lifetime and investigate the IFR preservation property under the formulation of m‐consecutive‐k‐out‐of‐n: F systems. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

On efficient algorithms for finding efficient salvo policies

We consider the salvo policy problem, in which there are k moments, called salvos, at which we can fire multiple missiles simultaneously at an incoming object. Each salvo is characterized by a probability p_i: the hit probability of a single missile. After each salvo, we can assess whether the incoming object is still active. If it is, we fire the missiles assigned to the next salvo. In the salvo policy problem, the goal is to assign at most n missiles to salvos in order to minimize the expected number of missiles used. We consider three problem versions. In Gould's version, we have to assign all n missiles to salvos. In the Big Bomb version, a cost of B is incurred when all salvo's are unsuccessful. Finally, we consider the Quota version in which the kill probability should exceed some quota Q. We discuss the computational complexity and the approximability of these problem versions. In particular, we show that Gould's version and the Big Bomb version admit pseudopolynomial time exact algorithms and fully polynomial time approximation schemes. We also present an iterative approximation algorithm for the Quota version, and show that a related problem is NP-complete.     

A search game with a protector

A classic problem in Search Theory is one in which a searcher allocates resources to the points of the integer interval [1, n] in an attempt to find an object which has been hidden in them using a known probability function. In this paper we consider a modification of this problem in which there is a protector who can also allocate resources to the points; allocating these resources makes it more difficult for the searcher to find an object. We model the situation as a two‐person non‐zero‐sum game so that we can take into account the fact that using resources can be costly. It is shown that this game has a unique Nash equilibrium when the searcher's probability of finding an object located at point i is of the form (1 − exp (−λ_ix_i)) exp (−μ_iy_i) when the searcher and protector allocate resources x_i and y_i respectively to point i. An algorithm to find this Nash equilibrium is given. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47:85–96, 2000     

Arrival and departure state distributions in the general bulk‐service queue

In this paper, we give an explicit relation between steady‐state probability distributions of the buffer occupancy at customer entrance and departure epochs, for the classical single‐server system G/G^[N]/1 with batch services and for the finite capacity case. The method relies on level‐crossing arguments. For the particular case of Poisson input, we also express the loss probability in terms of state probabilities at departure epochs, yielding probabilities observed by arriving customers. This work provides the “bulk queue” version of a result established by Burke, who stated the equality between probabilities at arrival and departure epochs for systems with “unit jumps.” © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 107–118, 1999     

Note: A geometrical method of solving certain games

One of the diagrammatic methods for solving two-person 2 × n matrix games can be extended to solve m × n games where each column of the matrix is a concave function of the row number. This gives a simple proof of a theorem of Benjamin and Goldman that such games have solutions involving no more than two consecutive strategies for the row player, and no more than two strategies for the column player. Two extensions are discussed. © 1994 John Wiley & Sons, Inc.     

A branch‐and‐cut algorithm for the nonpreemptive swapping problem

In the Swapping Problem (SP), we are given a complete graph, a set of object types, and a vehicle of unit capacity. An initial state specifies the object type currently located at each vertex (at most one type per vertex). A final state describes where these object types must be repositioned. In general, there exist several identical objects for a given object type, yielding multiple possible destinations for each object. The SP consists of finding a shortest vehicle route starting and ending at an arbitrary vertex, in such a way that each object is repositioned in its final state. This article exhibits some structural properties of optimal solutions and proposes a branch‐and‐cut algorithm based on a 0‐1 formulation of the problem. Computational results on random instances containing up to 200 vertices and eight object types are reported. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Scheduling jobs and maintenance activities on parallel machines

Most machine scheduling models assume that the machines are available all of the time. However, in most realistic situations, machines need to be maintained and hence may become unavailable during certain periods. In this paper, we study the problem of processing a set of n jobs on m parallel machines where each machine must be maintained once during the planning horizon. Our objective is to schedule jobs and maintenance activities so that the total weighted completion time of jobs is minimized. Two cases are studied in this paper. In the first case, there are sufficient resources so that different machines can be maintained simultaneously if necessary. In the second case, only one machine can be maintained at any given time. In this paper, we first show that, even when all jobs have the same weight, both cases of the problem are NP-hard. We then propose branch and bound algorithms based on the column generation approach for solving both cases of the problem. Our algorithms are capable of optimally solving medium sized problems within a reasonable computational time. We note that the general problem where at most j machines, 1 ≤ j ≤ m, can be maintained simultaneously, can be solved similarly by the column generation approach proposed in this paper. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 145–165, 2000     

IFR results for repairable systems

Consider a k-out-of-n system with independent repairable components. Assume that the repair and failure distributions are exponential with parameters {μ₁, ?,μ_n} and {λ₁, ?,λ_n}, respectively. In this article we show that if λ_i – μ_i = Δ for all i then the life distribution of the system is increasing failure rate (IFR).     

Parallel machine scheduling: A probabilistic analysis

The minimum makespan of the general parallel machine scheduling problem with m machines and n jobs is studied. As for a number of other important combinatorial problems, the theory of empirical processes proves to be a very elegant and powerful tool for the probabilistic analysis of the solution value. It is used in this paper to derive a scheduling constant θ such that, for random processing times, the minimum makespan almost surely grows as θn when n goes to infinity. Moreover, a thorough probabilistic analysis is performed on the difference between the minimum makespan and θn. Explicit expressions for the scheduling constant are given for an arbitrary number of unrelated machines with identically distributed processing times (with an increasing failure rate), and for an arbitrary number of uniform machines and generally distributed processing times. © 1996 John Wiley & Sons, Inc.